In a certain company, the formula for maximizing profits is

Couple of things:

Quadratic expression \(ax^2+bx+c\) reaches its extreme values when \(x=-\frac{b}{2a}\). When \(a>0\) extreme value is minimum value of \(ax^2+bx+c\) (maximum value is not limited) and when \(a<0\) extreme value is maximum value of \(ax^2+bx+c\) (minimum value is not limited).

You can look at this geometrically: \(y=ax^2+bx+c\) when graphed on XY plane gives parabola. When \(a>0\), the parabola opens upward and minimum value of \(ax^2+bx+c\) is y-coordinate of vertex, when \(a<0\), the parabola opens downward and maximum value of \(ax^2+bx+c\) is y-coordinate of vertex.

Examples:
Expression \(5x^2-10x+20\) reaches its minimum when \(x=-\frac{b}{2a}=-\frac{-10}{2*5}=1\), so minimum value is \(5x^2-10x+20=5*1^2-10*1+20=15\).

Expression \(-5x^2-10x+20\) reaches its maximum when \(x=-\frac{b}{2a}=-\frac{-10}{2*(-5)}=-1\), so maximum value is \(-5x^2-10x+20=-5*(-1)^2-10*(-1)+20=25\).

Back to the original question:

In a certain company, the formula for maximizing profits is P = -25x^2 + 7500x, where P is profit and x is the number of machines the company operates in its factory. What value for x will maximize P?

A) 10
B) 50
C) 150
D) 200
E) 300

\(P=-25x^2+7500x\) reaches its maximum (as \(a=-25<0\)) when \(x=-\frac{b}{2a}=-\frac{7500}{2*(-25)}=150\).

Answer: C.

As for the minimum value of P: minimum value of P is not limited (x increase to +infinity --> P decreases to -infinity).

Hope it helps.